Problem: Solve for $x$ and $y$ by deriving an expression for $x$ from the second equation, and substituting it back into the first equation. $\begin{align*}9x-5y &= -8 \\ 4x-5y &= 2\end{align*}$
Begin by moving the $y$ -term in the second equation to the right side of the equation. $4x = 5y+2$ Divide both sides by $4$ to isolate $x$ $x = {\dfrac{5}{4}y + \dfrac{1}{2}}$ Substitute this expression for $x$ in the first equation. $9({\dfrac{5}{4}y + \dfrac{1}{2}}) - 5y = -8$ $\dfrac{45}{4}y + \dfrac{9}{2} - 5y = -8$ Simplify by combining terms, then solve for $y$ $\dfrac{25}{4}y + \dfrac{9}{2} = -8$ $\dfrac{25}{4}y = -\dfrac{25}{2}$ $y = -2$ Substitute $-2$ for $y$ in the top equation. $9x-5( -2) = -8$ $9x+10 = -8$ $9x = -18$ $x = -2$ The solution is $\enspace x = -2, \enspace y = -2$.